BasicArgumentForms - Colorado State University

Basic Argument Forms

Name

Equivalence

Description

Modus Ponens

(p q) p q

if p then q; p; therefore q

Modus Tollens

(p q) ?q ?p

if p then q; not q; therefore not p

Hypothetical Syllogism

(p q) (q r) (p r)

if p then q; if q then r; therefore, if p then r

Disjunctive Syllogism

(p q) ?p q

Either p or q; not p; therefore, q

Constructive Dilemma

(p q) (r s) (p r) (q s)

if p then q; and if r then s; but either p or r; therefore either q or s

Destructive Dilemma

(p q) (r s) (?q ?s) (?p ?r)

if p then q; and if r then s; but either not q or not s; therefore either not p or not r

Simpli cation

(p q) p

p and q are true; therefore p is true

Conjunction

p, q (p q)

p and q are true separately; therefore they are true conjointly

Addition

p (p q)

p is true; therefore the disjunction (p or q) is true

Composition

(p q) (p r) (p (q r))

if p then q; and if p then r; therefore if p is true then q and r are true

De Morgan's eorem ( )

?(p q) (?p ?q)

e negation of (p and q) is equiv. to (not p or not q)

De Morgan's eorem ( ) Commutation ( ) Commutation ( ) Association ( ) Association ( ) Distribution ( ) Distribution ( ) Double Negation Transposition Material Implication Exportation Importation Tautology ( ) Tautology ( ) Tertium non datur (Law of Excluded Middle)

?(p q) (?p ?q)

e negation of (p or q) is equiv. to (not p and not q)

(p q) (q p)

(p or q) is equiv. to (q or p)

(p q) (q p)

(p and q) is equiv. to (q and p)

(p (q r)) p or (q or r) is equiv. to (p or q) or r ((p q) r)

(p (q r)) p and (q and r) is equiv. to (p and q) and r ((p q) r)

(p (q r)) p and (q or r) is equiv. to (p and q) or (p and r) ((p q) (p r))

(p (q r)) p or (q and r) is equiv. to (p or q) and (p or r) ((p q) (p r))

p ??p

p is equivalent to the negation of not p

(p q) (?q ?p)

if p then q is equiv. to if not q then not p

(p q) (?p q)

if p then q is equiv. to not p or q

((p q) r) from (if p and q are true then r is true) we can prove (if q is true (p (q r)) then r is true, if p is true)

(p (q r)) ((p q) r)

p (p p)

p is true is equiv. to p is true or p is true

p (p p)

p is true is equiv. to p is true and p is true

(p ?p)

p or not p is true

Source:

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